Model Theory and Differential Equations

这个项目集中在代数微分方程和模型理论。微分方程是现代数学及其在其他科学中的应用的核心。模型论是数理逻辑的一部分，其中研究被称为可定义集合的对象。可定义集合的概念是灵活的，随着人们在不同数学环境中的工作而变化。在微分代数领域，可定义集与微分方程紧密相连，提供了两个学科之间的联系。该项目计划利用这一联系来促进这两个领域的知识。该项目还寻求在其他数学领域的应用，主要是数论。这个研究项目解决的问题，在模型理论的领域与运营商。首先，研究者将推广他在j-函数的微分代数上的工作，以理解与各种志村簇相关的解析覆盖映射所满足的微分方程。这一系列的工作预计将有一些理论后果有关的特殊点结构，因为它没有在情况下的模曲线。该项目还将研究微分代数中的几个有限性结果;这一系列的工作预计将对数论和计算微分代数中的有效界产生影响。该项目旨在发展超简单和超稳定群的模型理论的各个方面，作为微分代数和差分代数群的最新工作的推广。该项目将研究微分闭域中分裂的强极小集的分类。这项工作旨在回答一个基本的问题，微分方程：什么是可能的结构给出的微分代数关系的解决方案，以一个固定的微分方程？最后，本计画的目的是将代数叶理与向量场的结果应用于微分代数几何。

This project centers on algebraic differential equations and model theory. Differential equations are at the heart of modern mathematics and its applications in other sciences. Model theory is a part of mathematical logic in which objects known as definable sets are studied. The notion of a definable set is flexible, changing as one works in different mathematical settings. In the area of differential algebra, definable sets are closely linked with differential equations, providing a connection between the two subjects. This project plans to exploit that link to advance the knowledge of both fields. The project also seeks applications in other areas of mathematics, primarily number theory.This research project addresses problems in the model theory of fields with operators. First, the investigator will generalize his work on the differential algebra of the j-function to understand the differential equations satisfied by analytic covering maps associated with various Shimura varieties. This line of work is expected to have number theoretic consequences related to special-point conjectures as it did in the case of modular curves. The project will also investigate several finiteness results in differential algebra; this line of work is expected to have consequences on effective bounds in number theory and in computational differential algebra. The project aims to develop various aspects of the model theory of supersimple and superstable groups as a generalization of recent work in differential algebraic and difference algebraic groups. The project will investigate the classification of disintegrated strongly minimal sets in differentially closed fields. This work seeks to answer a fundamental question about differential equations: what are the possible structures given by the differential algebraic relations between solutions to a fixed differential equation? Finally, this project aims to adapt and generalize results from algebraic foliations and vector fields for application in differential algebraic geometry.